Welcome to MathOverflow

Question

1

How can one prove that the convolution of f \in L^1 and g \in L^p is defined almost everywhere? Here f and g are measurable functions in R^n.

In general what techniques are there for showing some function is defined a.e and where should I look for them in analysis literature.

Background

This came up while I was reading some notes to understand harmonic analysis.

flag	asked Mar 17 2010 at 14:46 Nirman 19●1

1

This is explained in most good real analysis books. For example, Knapp's Basic real analysis. – Mariano Suárez-Alvarez Mar 17 2010 at 15:01

I would agree to this comment in the case p=1, but otherwise? – anton Mar 17 2010 at 15:09

(In particular, this follows from Minkowski's inequality, en.wikipedia.org/wiki/…) – Mariano Suárez-Alvarez Mar 17 2010 at 15:10

Correct me if I'm wrong,measure theory was several semesters ago-but isn't the simplest way to prove a function is defined almost everywhere is just to show the complement of it's support is finite? I don't remember if it's allowed to be countable,but since countable sets have measure 0 (VERY easy to prove) ,I'd be willing to bet the common definition allows for countable sets in the range where f(x)=0. Analysts,please feel free to jump in and correct me at any time here. – Andrew L Mar 18 2010 at 3:27

anton · Answer 1 · 2010-03-17 15:08:34Z UTC

1

If $g\in L^p$ for $1\le p\le\infty$, then there exists $C>0$ such that the function $g_1(x)=g(x){\bf 1}_A(x)$ is in $L^1$, where $A$ is the set where $|g(x)|>C$.That means that $g$ is the sum of an $L^1$-function plus a bounded function, so the convolution $f*g$ exists.

answered Mar 17 2010 at 15:08

anton
2,412●7●12

You can take any positive $C$ here; when $C=1$ the conclusion is particularly obvious. – Robin Chapman Mar 17 2010 at 15:28

Only if p is finite. If p is infinity, you have to take C to be equal to the infinity-norm of g. – anton Mar 17 2010 at 15:42

3

When $p=\infty$, I wouldn't use this method to prove that an $L^\infty$ function is the sum of an $L^1$ and an $L^\infty$ function :-) – Robin Chapman Mar 17 2010 at 18:56

Keenan Kidwell · Answer 2 · 2010-03-17 15:49:06Z UTC

Folland's real analysis text spends a lot of time on various basic $L^p$ inequalities including $L^p$ norms of convolutions. The material on convolutions is (I believe) in Chapter 8 which is titled ``Elements of Fourier Analysis," and the basic $L^p$ stuff is in Chapter 6. Incidentally, this is definitely not a research level question, and in fact, I would say it's a standard exercise (using Hölder\Minkowski\Fubini) in a first course in measure theory.

Defined Almost Everywhere

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Background

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.